Scale Effects on the Ballistic Penetration of Graphene Sheets

Carbon nanostructures are promising ballistic protection materials, due to their low density and excellent mechanical properties. Recent experimental and computational investigations on the behavior of graphene under impact conditions revealed exceptional energy absorption properties as well. However, the reported numerical and experimental values differ by an order of magnitude. In this work, we combined numerical and analytical modeling to address this issue. In the numerical part, we employed reactive molecular dynamics to carry out ballistic tests on single and double-layered graphene sheets. We used velocity values within the range tested in experiments. Our numerical and the experimental results were used to determine parameters for a scaling law, which is in good agreement with all experimental and simulation results. We find that the specific penetration energy decreases as the number of layers (N) increases, from ~25 MJ/kg for N=1 to ~0.26 MJ/kg as N goes to infinity. These scale effects explain the apparent discrepancy between simulations and experiments.

We attribute fluctuations in instantaneous cone velocity values to thermal effects. In order to employ conditions as close as possible to the experiments, we set the initial temperature of the system at T = 300 K. Because of this, the z position of every atom fluctuates during MD simulations. In the criteria employed, an atom is considered inside the deformation cone when its position is lower than a threshold (z < 12Å). If an atom is fluctuating down, its instantaneous position might be lower than the threshold, although its equilibrium position is not. The reverse might occur if it is fluctuating upwards. There is thus some uncertainty in our criteria. Note also that relative fluctuations decrease as impact velocities increase, which can be correlated to an increased ratio between kinetic and thermal energies (compare figures S1 and S2). Our finding that deformation cones propagate at constant velocities is corroborated by Haque et al. [1], that set initial temperatures to T = 1 K to avoid such effects.

DERIVATION OF THE EQUATIONS
According to Pugno [2,3], considering a collision generating large-sized fragments, the absorbed energy (E) is proportional to the volume dislocated during the collision process (V ) and the proportionality constant is close to the strength of the material (σ), so we have From this equation it is possible to derive the strength of an N-layered material as where ρ is the density of the target material and m is the mass of the affected region, which in our case can be calculated as where A f , N and t are respectively the damaged zone area, number of layers and thickness of the single layer. For the thickness we used the well-known graphene thickness t = 0.34 nm. So we have Introducing the η parameter where A p = πr 2 is the particle cross section area with r being the particle radius. So we end up with the equation presented in the paper is the well known specific penetration energy. Equation 7 allow us to relate the absorbed energy during collision with the strength of the target material, and to directly compare our d N results with the literature.
To do the scale analysis we followed the procedure presented in Pugno [4] relating the strength of the material with its structural size, here number of layers N where σ ∞ is the strength of the bulk material, while N c and N c are critical values to be determined.
So combining equations 6 and 8 we obtain the equation presented in the paper where d ∞ = σ ∞ /ηρ. From this equation it is possible to do a scale analysis considering the specific penetration energy instead of the strength of the material.

THE EFFECT OF PRE-TENSION IN GRAPHENE LAYERS DURING BALLISTIC PENETRATION
In this manuscript, we suggested that depositing graphene nanocoatings on low-density substrates (such as graphene sponges) would maximize specific penetration energy values.
It should be noted, however, that in such a system graphene should not be pre-tensioned at the onset of impact. If that is the case, the estimated values of specific penetration energy could differ from those predicted here. Xia et al. showed that the shape of the graphene membrane used in a ballistic penetration test influences stress distribution, which in turn influences the specific penetration energy (SPE) [5]. As the absence of pre-tension is very likely to lead to different stress distribution at impact, it might also lead to modified values of specific penetration energy. Note, however, that the variations found by Xia et al. never exceeded 20%. We therefore expect our main conclusions to remain valid.